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Rapid Acceleration In an ultra-fast regime—applicable to unstable particles—acceleration is completed in a few to a few tens of turns Magnetic field cannot be ramped Magnetic field cannot be ramped RF parameters are fixed—no phase/voltage compensation is feasible RF parameters are fixed—no phase/voltage compensation is feasible operate at or near the rf crest operate at or near the rf crest Fixed-field lattices have been developed which can contain up to a factor of 4 change in energy; typical is a factor of 2-3 There are three main types of fixed field lattices under development: Conventional Recirculating Linear Accelerators (RLAs) Scaling FFAG (Fixed Field Alternating Gradient) Linear, nonscaling FFAG

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Current Baseline: Recirculating Linacs A Recirculating Linac Accelerator (RLA) consists of two opposing linacs connected by separate, fixed-field arcs for each acceleration turn In Muon Acceleration for a Neutrino Factory: The RLAs only support ONLY 4 acceleration turns The RLAs only support ONLY 4 acceleration turns due to the passive switchyard which must switch beam into the appropriate arc on each acceleration turn and the large momentum spreads and beam sizes involved. due to the passive switchyard which must switch beam into the appropriate arc on each acceleration turn and the large momentum spreads and beam sizes involved.  2-3 GeV of rf is required per turn (NOT DISTRIBUTED)  2-3 GeV of rf is required per turn (NOT DISTRIBUTED) Again to enable beam separation and switching to separate arcs Again to enable beam separation and switching to separate arcs Advantage of the RLA Beam arrival time or M56 matching to the rf is independently controlled in each return arc, no rf gymnastics are involved; I.e. single-frequency, high-Q rf system is used. RLAs comprise about 1/3 the cost of the U.S. Neutrino Factory

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Mulit-GeV FFAGs: Motivation Ionization cooling is based on acceleration Ionization cooling is based on acceleration - (deacceleration of all momenum components then longitudinal reacceleration) THERE is a STRONG argument to let the accelerator do the bulk of the LONGITUDINAL AND TRANSVERSE COOLING (adiabatic cooling). The storage ring can accept ~  4%  p/p @20 GeV If acceleration is completely linear, so that absolute momentum spread is preserved, @ ~400 MeV  p/p =  200%  p/p =  200% implying no longitudinal cooling. implying no longitudinal cooling. (Upstream Linear channels for TRANSVERSE Cooling currently accept a maximum of  22% for the solenoidal sFOFO and -22% to +50% for quadrupoles). The Linac/RLA has been the showstopper in this argument The Linac/RLA has been the showstopper in this argument

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Mulit-GeV FFAGs for a Neutrino Factory or Muon Collider Lattices have been developed which, practically, support up to a factor of 4 change in energy, or Lattices have been developed which, practically, support up to a factor of 4 change in energy, or almost unlimited momentum-spread acceptance, which has immediate consequences on the degree of ionisation cooling required almost unlimited momentum-spread acceptance, which has immediate consequences on the degree of ionisation cooling required Practical, technical considerations (magnet apertures, mainly, and rf voltage) have resulted in a chain of FFAGs with a factor of 2 change in energy Practical, technical considerations (magnet apertures, mainly, and rf voltage) have resulted in a chain of FFAGs with a factor of 2 change in energy 2.5 -5 GeV 5-10 GeV 10-20 GeV Currently proposal, U.S. scenario

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Scaling FFAGs (radial sector): The B field and orbit are constructed such that the B field scales with radius/momentum such that the optics remain constant as a function of momentum. Scaling machines display almost unlimited momentum acceptance, but a more restricted transverse acceptance than linear nonscaling FFAGs and more complex magnets. KEK, Nufact02, London

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Perk of Rapid Acceleration* Freedom to cross betatron resonances: optics can change slowly with energy optics can change slowly with energy allows lattice to be constructed from linear magnetic elements (dipoles and quadrupoles only) allows lattice to be constructed from linear magnetic elements (dipoles and quadrupoles only) This is the basic concept for a linear non-scaling FFAG * In muon machines acceleration is completed in milliseconds or tens of milliseconds

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Linear non-scaling FFAGs: Transverse acceptance: “unlimited” due to linear magnetic elements “unlimited” due to linear magnetic elements Large horizontal magnet aperture Large horizontal magnet aperture General characteristic of fixed-field acceleration General characteristic of fixed-field acceleration Orbit changes as a function of momentum: beam travels from the inside of the ring to the outside Orbit changes as a function of momentum: beam travels from the inside of the ring to the outside Momentum Acceptance: FODO optics: FODO optics: Large range in momentum acceptance: Large range in momentum acceptance: defined by lower and upper limits of stability defined by lower and upper limits of stability Limits depend on FODO cell parameters Limits depend on FODO cell parameters Triplet, doublet (dual-plane focusing) optics: Triplet, doublet (dual-plane focusing) optics: Too achromatic; small momentum acceptance to achieve horizontal+vertical foci. Too achromatic; small momentum acceptance to achieve horizontal+vertical foci.

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Phase advance in a linear non-scaling FFAG Stable range as a function of momentum Stable range as a function of momentum Lower limit: Lower limit: Given simply and approximately by thin-lens equations for FODO optics Given simply and approximately by thin-lens equations for FODO optics Upper limit: Upper limit: No upper limit in thin-lens approximation No upper limit in thin-lens approximation Have to use thick lens model Have to use thick lens model

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In the thin-lens approximation, the phase advance, , is given by In the thin-lens approximation, the phase advance, , is given by with f being the focal length of ½ quadrupole and L the length of a half cell from quadrupole center to center In equation (3), B’ is the quadrupole gradient in T/m and p is the momentum in GeV/c. Selecting  = 90  at p 0, the reference momentum implies the following:

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Differentiating the above equation gives the dependence of phase advance on momentum Differentiating the above equation gives the dependence of phase advance on momentum There is a low-momentum cut-off, but at large p, the phase advance varies more and more slowly, as 1/p 2, and there is no effective high-momentum cut-off in the thin-lens approximation. A high-momentum stability limit is observed in the thick lens representation

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Using thin-lens solutions, the peak beta function for a FODO cell is given by: In the above equation can only be set to locally (at ~76  ), but this does not guarantee stability in the beta function over a large range in momentum. The only approach that minimizes over a broad spectrum is to let. In the above equation (7), (  2 -  - 1) can only be set to 0 locally (at ~76  ), but this does not guarantee stability in the beta function over a large range in momentum. The only approach that minimizes d  max /dp over a broad spectrum is to let L approach 0.

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Travails of Rapid Fixed Field Acceleration  A pathology of fixed-field acceleration in recirculating-beam accelerators (for single, not multiple arcs) is that the particle beam transits the radial aperture  The orbit change is significant and leads to non-isochronism, or a lack of synchronism with the accelerating rf  The result is an unavoidable phase slippage of the beam particles relative to the rf waveform and eventual loss of net acceleration with The lattice completely determining the change in circulation time (for ultra relativistic particles) The lattice completely determining the change in circulation time (for ultra relativistic particles) The rf frequency determining the phase slippage which accumulates on a per turn basis: The rf frequency determining the phase slippage which accumulates on a per turn basis:

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Linear Dispersion in thin-lens FODO optics  Dispersion can be expressed in standard thin-lens matrix formalism.  At the symmetry points of the FODO cell the slope of optical parameters is zero, and correspond to points of maximum and minimum dispersion. For horizontal dispersion, the center of the vertically-focusing element is a minimum and horizontally-focusing element is a maximum. element is a maximum.

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Thin lens matrix solutions for different dipole options in a FODO  The transfer matrix for a dipole field centered in the drift between focusing elements: 1/2F-drift-1/2D is:  For a dipole field centered in the vertically- in the vertically- focusing element: focusing element:

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Dispersion and dipole location  Dispersion solution for conventional FODO  Dispersion solution for the dipole field located in the vertically- focusing element—clearly reduced

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“Optimum” Minimum Momentum-compaction lattices for nonscaling FFAGs  The optimum lattices are strictly FODO-based, with two candidates: Combined Function (CF) FODO Combined Function (CF) FODO Horizontally-focusing quadrupole, and combined function horizontally- defocussing magnetHorizontally-focusing quadrupole, and combined function horizontally- defocussing magnet The rf drift is provided between the quadrupole and CF elementThe rf drift is provided between the quadrupole and CF element FODO – like triplet FODO – like triplet The horizontally-focusing quadrupole is split and the rf drift is inserted between the two halves.The horizontally-focusing quadrupole is split and the rf drift is inserted between the two halves. The magnet spacing between the quadrupole and the CF magnet is much reduced.The magnet spacing between the quadrupole and the CF magnet is much reduced.  All optical units have reflective symmetry, implying  ring  =  cell  =  1/2 cell Special insertions for rf, extraction, injection, etc. have not been successful Special insertions for rf, extraction, injection, etc. have not been successful

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Transfer matrices for triplet (FDF) FODO cells  For an rf drift inserted at the center of the horizontally-focusing quadrupole: - Note that the half cell contains only half the rf drift, hence the added drift matrix is L rf /2, rather than the half-cell length as in the FODO cell case. Where D, the distance between quadrupole centers, L rf /2 replaces the half-cell length

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Dispersion function for modified FODO; triplet quadrupole configuration  The combined focal length, f*, is the general result for a doublet quadrupole lens system.  With the rf drift placed at the center of the horizontally focusing element, the differences between them and from the FODO cell are not immediately obvious we unless we explore the possible values for f 1 and f 2.

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Limit of stability  One can solve for focal lengths in the limits of stability and use their relative scaling over the entire acceleration range as a basis for comparison between FODO cell configurations.  In the presence of no bend, 90 degrees of phase advance across a half cell represents the limit of stability for FODO-like optics (single minimum). This implies for a initial position on the x axis (x,x’=0), that its position will be 0 (x=0,x’) after a half-cell transformation, conversely for the y plane

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Closed orbit in the limit of stability  These are the only closed orbits at the limits of stability (180  ):  There is no “area” transmitted, beta functions go to infinity,  0, phase space is a line. [x,0] [0,y’] y’ y y [0,x’] [y,0] [-x,0] [0,-y’]

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Solutions for the limit of stability  For CF or separated FODO cells:  In the modified FDF FODO:

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Final Comparison, CF vs. modified FODO  One can now compare the decrease in dispersion in the limit of stability (using L ~1.5 D for the rf drifts, magnet spacing and lengths we use in actual designs). FODO FDF FODO FDF  At this point, one invokes scaling in focal length and bend angle to generalize conclusions over the entire momentum range in the thin-lens approximation.

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High-frequency (~200 MHz) RF acceleration In a nonscaling linear FFAG, the orbital pathlength, or  T, is parabolic with energy. At high-frequency,  100 MHz, the accumulated phase slip is significant after a few turns, In a nonscaling linear FFAG, the orbital pathlength, or  T, is parabolic with energy. At high-frequency,  100 MHz, the accumulated phase slip is significant after a few turns, The phase-slip can reverse twice with an implied potential for the beam’s arrival time to cross the crest three times, given the appropriate choice of starting phase and frequency The phase-slip can reverse twice with an implied potential for the beam’s arrival time to cross the crest three times, given the appropriate choice of starting phase and frequency harmonic of rf = point of phase reversal

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We know from the parabolic dependency of the circumference the explicit dependence of  (and therefore  ) on  must be We know from the parabolic dependency of the circumference the explicit dependence of  (and therefore  ) on  must be This implies This implies Where the coefficients correspond to momentum compaction at the lower and upper momentum, respectively and  is taken as the momentum offset to the central energy, or bottom of the parabola.  coefficiencts are kept postive so that a negative momentum compaction is clearer Where the coefficients correspond to momentum compaction at the lower and upper momentum, respectively and  is taken as the momentum offset to the central energy, or bottom of the parabola.  coefficiencts are kept postive so that a negative momentum compaction is clearer

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This implies This implies One can study the first coefficient,  0, to determine the behavior of  C as a function of free parameters One can study the first coefficient,  0, to determine the behavior of  C as a function of free parameters One can study the behavior at one momentum; the lower limit of stability, for example One can study the behavior at one momentum; the lower limit of stability, for example which makes the problem substantially easier to parameterize which makes the problem substantially easier to parameterize

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Circumference change in the thin-lens model In the thin-lens model, the total change in circumference (and therefore the total phase slip, or  T) can be estimated from the solutions found for dispersion about the central orbit (  =0). (The contributing term in  is managed by symmetrizing the parabola.) In the thin-lens model, the total change in circumference (and therefore the total phase slip, or  T) can be estimated from the solutions found for dispersion about the central orbit (  =0). (The contributing term in  is managed by symmetrizing the parabola.) For a periodic ring In the thin lens,  is linear across the half cell: D is the half cell length and s  D is the distance from the center of the CF quadrupole

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Integrating over the length of the CF quadrupole, l B, (for a half cell) and noting that in the thin-lens limit of stability  min =0 Integrating over the length of the CF quadrupole, l B, (for a half cell) and noting that in the thin-lens limit of stability  min =0

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For cost, you desire N small, so cost and time of flight/circumference change are opposing conditions. The best you can do is try and minimize l B and N simultaneously For cost, you desire N small, so cost and time of flight/circumference change are opposing conditions. The best you can do is try and minimize l B and N simultaneously Further near the acceleration range we have chosen, 2:1, the dependence goes as the square of the range. The original designs were  3, with either the  C or the circumference a factor of 9 larger. This represents the biggest factor in the phase-slip profile/design. Further near the acceleration range we have chosen, 2:1, the dependence goes as the square of the range. The original designs were  3, with either the  C or the circumference a factor of 9 larger. This represents the biggest factor in the phase-slip profile/design.

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One can immediately draw certain conclusions: One can immediately draw certain conclusions:

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Scaling with energy/momentum lower energy rings* Naively one would hope that circumference would scale with momentum. However, we know that  T or  C must be held at a certain value for successful acceleration. If  C is set or scaled relative to the High Energy Ring (HER), then a Low Energy Ring (LER) would follow: Naively one would hope that circumference would scale with momentum. However, we know that  T or  C must be held at a certain value for successful acceleration. If  C is set or scaled relative to the High Energy Ring (HER), then a Low Energy Ring (LER) would follow: *see FFAG workshop, TRIUMF, April, 2004, C. Johnstone, “Performance Criteria and Optimization of FFAG lattices for derivations

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Scaling Law: Phase-slip/cell If you want is  C/N to remain constant (phase-slip per cell) If you want is  C/N to remain constant (phase-slip per cell) The scaling law is then approximately: The scaling law is then approximately: This is somewhat optimistic because you are simply keeping the number of turns, and  T ~ constant. This is somewhat optimistic because you are simply keeping the number of turns, and  T ~ constant. For our rings this implies the 2.5-5 GeV ring is only ~60% the size of the 10-20 GeV ring. S. Berg’s optimizer finds 80% so this is fairly close for an approximate description For our rings this implies the 2.5-5 GeV ring is only ~60% the size of the 10-20 GeV ring. S. Berg’s optimizer finds 80% so this is fairly close for an approximate description

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Asynchronous Acceleration The number of phase reversals (points of sychronicity with the rf) = number of fixed points in the Hamiltonian The number of phase reversals (points of sychronicity with the rf) = number of fixed points in the Hamiltonian Scaling FFAGs with a linear dependence of pathlength on momentum have 1 fixed point Scaling FFAGs with a linear dependence of pathlength on momentum have 1 fixed point Linear nonscaling FFAGs with a quadratic pathlength dependence have 2 Linear nonscaling FFAGs with a quadratic pathlength dependence have 2 The number of fixed points = number of asynchronous modes of acceleration The number of fixed points = number of asynchronous modes of acceleration